How do you find the integral of int tan^n(x)∫tann(x) if n is an integer?
3 Answers
See the explanation section below.
Explanation:
For
= int tan^(n-2) (sec^2x - 1) dx=∫tann−2(sec2x−1)dx
= int tan^(n-2) sec^2x dx - int tan^(n-2) x dx=∫tann−2sec2xdx−∫tann−2xdx
The first integral is
The second integral is of the same form as the first. So, repaet the process until you arrive at
(for
or
(for
The general formula is too difficult for elementary calculus classes.
You have to determine these the hard way, sorry. I will do the first three.
Multiply by
= int (secxtanx)/secxdx=∫secxtanxsecxdx
Let:
=> int 1/udu⇒∫1udu
= ln|u|=ln|u|
= color(blue)(ln|secx| + C)=ln|secx|+C
Ironically, the second degree
Use trig identity and follow up:
= int sec^2x - 1dx=∫sec2x−1dx
= color(blue)(tanx - x + C)=tanx−x+C
And the third:
Split into familiar first and second-degree
= int tan^2x tanx dx=∫tan2xtanxdx
Use trig identity:
= int tanx(sec^2x - 1)dx=∫tanx(sec2x−1)dx
= inttanxsec^2x - tanxdx=∫tanxsec2x−tanxdx
Split into familiar
= int secxtanx*secx - tanxdx=∫secxtanx⋅secx−tanxdx
Let:
=> int udu - inttanxdx⇒∫udu−∫tanxdx
Recall the first-degree
= color(blue)(sec^2x/2 - ln|secx| + C)=sec2x2−ln|secx|+C
Hopefully that gave you the general techniques you can use to approach higher degree problems.
See the explanation.
Explanation:
Maybe this helps:
Let
since:
Dividing polynomial
and reminder
which sign depends on
Hence,
When
And hence:
Hence,